## Abstract

We present a new approximation algorithm for the treewidth problem which finds an upper bound on the treewidth and constructs a corresponding tree de-composition as well. Our algorithm is a faster variation of Reed’s classical algorithm. For the benefit of the reader, and to be able to compare these two algorithms, we start with a detailed time analysis of Reed’s algorithm. We fill in many details that have been omitted in Reed’s paper. Computing tree decompositions parameterized by the treewidth k is fixed parameter tractable (FPT), meaning that there are algorithms running in time O(f(k)g(n)) where f is a computable function, and g(n) is polynomial in n, where n is the number of vertices. An analysis of Reed’s algorithm shows f(k) = 2^{O(klogk)} and g(n) = n log n for a 5-approximation. Reed simply claims time O(n log n) for bounded k for his constant factor approximation algorithm, but the bound of 2^{Ω(klogk)}n log n is well known. From a practical point of view, we notice that the time of Reed’s algorithm also contains a term of O(k^{2}2^{24k}n log n), which for small k is much worse than the asymptotically leading term of 2^{O(klogk)}n log n. We analyze f(k) more precisely, because the purpose of this paper is to improve the running times for all reasonably small values of k. Our algorithm runs in O(f(k)n log n) too, but with a much smaller dependence on k. In our case, f(k) = 2^{O(k)}. This algorithm is simple and fast, especially for small values of k. We should mention that Bodlaender et al. [2016] have an algorithm with a linear dependence on n, and Korhonen [2021] obtains the much better approximation ratio of 2, while the current paper achieves a better dependence on k.

Original language | English (US) |
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Pages (from-to) | 257-282 |

Number of pages | 26 |

Journal | Journal of Graph Algorithms and Applications |

Volume | 26 |

Issue number | 2 |

DOIs | |

State | Published - 2022 |

## All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- General Computer Science
- Computer Science Applications
- Geometry and Topology
- Computational Theory and Mathematics